Interpolation and Divisibility of Meromorphic Matrix Functions · rational matrix functions; numerous interpolation problems for this class of functions and their various applications,

Ž .JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 199, 592]619 1996ARTICLE NO. 0163

Interpolation and Divisibilityof Meromorphic Matrix Functions

M. Rakowski*

Department of Mathematics, The Ohio State Uni ersity, Columbus, Ohio 43210

and

L. Rodman†

Department of Mathematics, College of William and Mary, Williamsburg, Virginia23187-8795

Communicated by J. W. Helton

Received March 2, 1994

We construct a meromorphic matrix function with given spectral data in theform of null and pole functions, coupling matrix, and left annihilator at every pointin the domain of definition of the function. Based on these results, descriptions aregiven of minimal divisibility of meromorphic matrix functions in terms of restric-

Ž .tions appropriately understood of their spectral data. Q 1996 Academic Press, Inc.

1. INTRODUCTION

Ž .Let V be a domain in the complex plane C, and let MM V be the setm= nof all m = n matrices whose entries are meromorphic functions defined inV. In this paper we solve the interpolation problem of constructing a

Ž .meromorphic matrix function W g MM V with given spectral data in them= nform of the null and pole functions, coupling matrix, and left annihilator atevery point in V. These results are further applied to obtain a description

Žof divisors of meromorphic matrix functions in terms of restrictions ap-.propriately understood of their spectral data.

In the next section we recall the construction of spectral data forŽ . w xW g MM V , which was introduced and studied in 2 ; in turn, it is basedm= n

w xon earlier work on rational matrix functions 8]10 . A full exposition of the

* Partially supported by NSF Grant DMS-9302706.† Partially supported by NSF Grant DMS-9123841.

592

0022-247Xr96 $18.00Copyright Q 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

MATRIX FUNCTIONS 593

theory of spectral data and interpolation problems for regular rationalw xmatrix functions is found in 6 .

The construction of spectral data given in Section 2 leads to an abstractconcept of admissible quadruples which is introduced and studied inSection 3. In Section 4 we state and prove our main interpolation result. InSection 5 we apply this result to describe classes of minimal divisors ofmeromorphic matrix functions. These problems in the framework of regu-

Žlar i.e., of square size and having determinant not identically equal to. w xzero meromorphic matrix functions have been addressed in 4 . Other

interpolation problems for analytic and meromorphic regular matrix func-w xtions have been studied in 31, 26, 27 .

The definitions and results in this paper are stated for V a domain in C.All of them can be extended to the situation when V is assumed to be a

� 4 � 4domain in the Riemann sphere C j ` such that V / C j ` . The case� 4 Ž .V s C j ` which we will not consider in this paper corresponds to

rational matrix functions; numerous interpolation problems for this classof functions and their various applications, especially in systems andcontrol, have been extensively studied recently. We mention only the bookw x w x6 and the papers 1, 13, 20, 3, 8 on this subject; many important

w xapplications, in particular to the HH control, are found in 6, 11, 12, 14, 28 .`

ŽThe following well-known diagonalization result also called the.Smith]McMillan form is fundamental for the development of material in

this paper:

Ž . Ž .PROPOSITION 1.1. Let W z g MM V where z g V is the independentm= nŽ . Ž .äriable. Then there exist analytic and inërtible on V matrix functions E z

Ž .and F z of sizes m = m and n = n, respecti ely, such that

D z 0Ž .E z W z F z s .Ž . Ž . Ž .0 0

Ž . Ž Ž . Ž ..Here D z s diag d z , . . . , d z is a diagonal matrix function, where1 kŽ . Ž . Ž .d z k 0, . . . , d z k 0 are meromorphic on V scalar functions such that1 kŽ .Ž Ž ..y1d z d z is analytic in V, for i s 1, . . . , k y 1. Moreoër, the integeriq1 i

Ž Ž .. Ž . Ž .k 0 F k F min m, n is determined uniquely by W z , and d z areidetermined uniquely up to multiplication by an analytic nowhere zero function.

w xProposition 1.1 goes back to 36 , where it was proved for analytic matrixw xfunctions. A full proof of Proposition 1.1 is also found in 25 .

Ž .The number k in Proposition 1.1 is called the normal rank of W z . AŽ Ž ..point z g V is called regular for W z if z is neither a zero nor a pole0 0

Ž . Ž .of any of the scalar functions d z , . . . , d z ; otherwise z is called1 k 0singular.

RAKOWSKI AND RODMAN594

Ž .A local version whose proof is easily obtained from Proposition 1.1 ofthe Smith]McMillan form is also useful:

Ž . Ž .PROPOSITION 1.2. Let W z g MM V , and let z g V. Then there existm= n 0Ž . Ž .analytic and inërtible at z matrix functions E z and F z of sizes m = m0 z z0 0

and n = n, respecti ely, such that

D z 0Ž .z0E z W z F z s ,Ž . Ž . Ž .z z0 0 0 0

Ž . ŽŽ .a1 Ž .ak .where D z s diag z y z , . . . , z y z . Moreoër, the integers az 0 0 10Ž .G ??? G a are uniquely determined by W z and by z .k 0

Ž .The positive integers if any among a , . . . , a are called the null1 kŽ .multiplicities of W z at z , and the number of positive integers are called0

the geometric multiplicity of the zero of W at z . The absolute values of the0Ž .negative integers if any among the a ’s are called the pole multiplicities ofj

Ž .W z at z and the number of negative a ’s is called the geometric0 iŽ .multiplicity of the pole of W at z . A point z g V is called a pole of W z0 0

Ž .if z is a pole of at least one entry of W z , or, equivalently, if there is a0Ž . Ž .pole multiplicity of W z at z . A point z g V is called a zero of W z if0 0

Ž .there is a null multiplicity of W z at z .0We conclude the introduction with some matrix notation that is used in

the paper. The set of all complex m = n matrices is denoted Cm= n; the setof columns Cm= 1 is often abbreviated to Cm. Block diagonal and blockcolumn matrices are denoted as

A 0 ??? 01

0 A ??? 02diag A , . . . , A s ;Ž . . . .1 k . . .. . .

0 0 ??? Ak

Z Z1 k

Z Z2 ky1k 1col Z s ; col Z s .Ž . Ž .. .i iis1 isk. .. .Z Zk 1

Ž . Ž .Finally, s A stands for the spectrum the set of eigenvalues of thecomplex matrix A.


2. THE SPECTRAL DATA

Ž . ŽThroughout this section, we fix a function W g MM V here andm= n.elsewhere in this paper V is a fixed domain in C . We describe in this

w xsection the spectral data of W. The description will be based on 2 ; anŽ .analogous description restricted to rational matrix functions is found in

w x9 .We begin with pole pairs which describe the local pole structure of W.

Ž .Let l g V be a pole of W. A meromorphic in a neighborhood of lŽ .m = 1 matrix-valued function c is called a right pole function for W at l

if c has a pole at l, and c s Wf for some vector function f which isanalytic in a neighborhood of l and does not vanish at l. The multiplicity

Ž .of the pole of c at l is called the order of c as a right pole function forW at l.

Ž w xIt will be convenient to use the following notion of orthogonality see 9. Ž .for more details . Let f z be a meromorphic scalar function in a

< < < < yhneighborhood of l g C. We let f s 0 if f s 0, and f s e ifzsl zslh˜ ˜Ž . Ž . Ž . Ž .f / 0, where h is the integer such that f z s z y l f z , and f z is

analytic and nonzero at l. For a column or row vector-value meromorphicŽ . Ž Ž ..n Ž . Ž Ž . Ž ..function f z s col f z or f z s f z ??? f z definej js1 1 n

5 5 < < < <f s max f , . . . , f .� 4zsl zsl zsl1 n

Ž . Ž .Two vector-valued meromorphic functions f z and c z are calledorthogonal at l if

5 5 5 5 5 5� 4af q bc s max af , bczsl zsl zsl

Ž .for all scalar meromorphic in a neighborhood of l functions a and b.� 4A set c , c , . . . , c of pole functions for W at l is said to be canonical1 2 h

if the functions c , c , . . . , c are orthogonal at l and, for each pole1 2 h

function c for W at l, there exist scalar functions f , f , . . . , f that are1 2 h

analytic in a neighborhood of l and such that the function c y f c y1 1f c q ??? qf c is analytic as well in a neighborhood of l. Let2 2 h h

� 4c , c , . . . , c be a canonical set of pole functions for W at l. If1 2 h

`jyk iz y l c zŽ . Ž .Ý i j

js0

with c / 0 is the Laurent expansion of c in a neighborhood of l, leti0 i

C s c c ??? ci i0 i1 iŽk y1.i

Ž .i s 1, 2, . . . , h . Let

C , A s C C ??? C , diag J l , . . . , J l ,Ž . Ž . Ž .Ž .ž /l l 1 2 h k k1 h


Ž .where J l is the upper triangular k = k Jordan block with l on thekŽ . Ž .diagonal. Any pair of matrices C, A which is right-similar to C , A isl l

Ž . Ž .called a right pole pair for W at l. Here right-similarity of C , A andl l

Ž .C, A means that there exists a nonsingular matrix S such that C s C Sl

and A s Sy1A S.l

More generally, let s be any subset in V such that W has only a finitenumber of poles in s , and let l , l , . . . , l be the distinct poles in W in1 2 k

Ž . Ž .s . Let C , A be a pole pair for W at l i s 1, 2, . . . , k . Any pair ofi i imatrices right-similar to the pair

w xC C . . . C , diag A , A , . . . , AŽ .Ž .1 2 k 1 2 k

Ž .is called a right pole pair for W over s . It follows from the constructionthat a pole pair for W over s exists whenever W has a finite number of

Ž .poles in s , and that a pole pair C, A for W over s is observable, i.e.,` Ž j. � 4F Ker CA s 0 . The uniqueness of a pole pair for W over s isjs0

characterized in the following proposition.

PROPOSITION 2.1. Suppose a meromorphic matrix function W has a finite˜ ˜Ž . Ž .number of poles in s , and let C, A be a s-pole pair of W. Then C, A is a

right pole pair for W oër s if and only if

˜ ˜ y1C s CS and A s S AS 2.1Ž .

for some nonsingular matrix S.

The proof of this proposition, as well as of several other statements inthis section, will be omitted. All the omitted proofs can be found either inw x w2 , or by a suitable adaptation of proofs given in the rational case in 9,

x10 .Ž .We note that since the pair C, A in Proposition 2.1 is observable,

Ž . Žequalities 2.1 determine the matrix S uniquely see, e.g., proof ofw x.Theorem 7.14 in 22 .

Right pole pairs of a function W can be equivalently defined as follows.

PROPOSITION 2.2. Let W be a meromorphic matrix function with a finiteŽ .number of poles in s and let C, A be an obser able pair of matrices with

Ž . Ž .s A : s . Then C, A is a right pole pair for W oër s if and only if thereŽ .exists a matrix B such that the pair A, B is controllable and the function

y1W z y C zI y A B 2.2Ž . Ž . Ž .

is analytic on s .

Ž .Recall that a pair of matrices A, B , where is p = p and B is p = q, is` Ž j . pcalled controllable if Ý Im A B s C .js0


Next, we pass to the construction of left null pairs. Let s be a subset ofV. We will denote by MM the field of scalar functions which are meromor-s

Ž .phic in some open set containing s , and by MM s the MM -vector spacem= n s

of m = n matrix-valued functions which are meromorphic in an open setŽ .containing s . The concept of orthogonality of MM -subspaces in MM ss m=n

carries over to this setting.Ž .Let W g MM s be a function, and letm= n

W 0 l s f g MM s : fW s 0 .� 4Ž .1=m

Ž . Ž . 0 lChoose a subspace L over MM such that MM s s W q L ands 1=mW 0 l, L are orthogonal at a point l g s . A function f g L is called a leftnull function for W at l g s if f is analytic and does not vanish at l andŽ .Ž .fW l s 0. The order to which fW vanishes at l is called the order off as a left null function for W at l. A set of left null functions� 4f , f , . . . , f : L for W at l is said to be canonical if1 2 k

Ž .i f , f , . . . , f are orthogonal at l;1 2 k

Ž .ii the sum of orders of f , f , . . . , f is maximal.1 2 k

It can be shown using Proposition 1.2 that the number and orders offunctions in a canonical set of left null functions of W at l do not dependon the choice of L. These orders coincide with the null multiplicities of Wat l.

Ž . 0 l 0 lSuppose now that MM s s W q L, and the subspaces W and L1=mare orthogonal at each zero of W in s . Suppose in addition that W has afinite number of zeros in s , say, l , l , . . . , l . For a fixed zero l of W in1 2 r i

Ž .s choose from functions in L a canonical set of left null functions� 4 Žf , f , . . . , f for W at l f depend, of course, on l ; we suppress this1 2 k i j i

. Ž .dependence in our notation . If f j s 1, . . . , k is a null function of orderjh and has the Taylor expansion at lj i

`ly1

f z s z y l f ,Ž . Ž .Ýj i j , lls1

Ž Ž . Ž . Ž .. Ž Ž .hpy1 .klet A s diag J l , J l , . . . , J l and B s col col f .i h i h i h i i p, h yj js0 ps11 2 k pŽ Ž . Ž . r .Any pair of matrices left-similar to the pair diag A , . . . , A , col B1 r j js1

Ž .is called a left null pair for W over s . Here left-similarity of pairs A, BŽ .and G, H means that there exists a nonsingular matrix S such that

G s SASy1 and H s SB. A left null pair for W over s is necessarilycontrollable.

It follows from the definition that for each given left null pair for WŽ .over s there exists a subspace L of MM s such that L is orthogonal to1=m

0 l Ž . Ž .W on s A and the pair A, B is similar to a pair constructed from


Taylor coefficients of functions in L. We will call L a subspace associatedŽ .with the pair A, B .

The nonuniqueness of left null pairs can be characterized as follows.

w x Ž .PROPOSITION 2.3. 2 . Let A, B be a left null pair at l g C of anm = n matrix function W which is meromorphic in a neighborhood of l.Suppose the normal rank of W is m y k , and let h be the largest nullmultiplicity of W at l. Choose functions p , p , . . . , p g W 0 l such that1 2 k

Ž . Ž . Ž .p , p , . . . , p are analytic at l and p l , p l , . . . , p l are linearly1 2 k 1 2 k

independent. If

`ly1z y l pŽ .Ý i l

ls1

Ž .is the Taylor expansion of p at l i s 1, 2, . . . , k , let A si k

Ž Ž . Ž .. Ž Ž . .diag J l , . . . , J l the block J l appears k times in A and B sk h h h k k

Ž Ž .1 .k Ž .col col p . Then the pair of matrices G, H is a left null pair for Wil lsh is1� 4oër l if and only if the pairs

A 0 B G 0 H, , and , 2.3Ž .0 A B 0 A Bž / ž /k k k k

are left similar.

Ž .As before, let s be a subset in V. The left null-pole subspace of Wover s is the complex vector space

�S W s Wf : f is an n = 1 vector function analyticŽ .s

4in an open set containing s .

The characterization of null-pole subspaces of regular rational matrixw x Ž w x.functions in 7 see also 6 , which has been generalized to the nonregular

w xcase in 10 , carried over to meromorphic functions.

Ž .PROPOSITION 2.4. Let W g MM V be a function with normal rankm= nŽ .m y k that has a finite number of poles and zeros in s : V. Let P g MM sk=m

Ž . Ž .be such that P z W z s 0 for each z in an open set containing s and theŽ .normal rank of P is k. Choose a right pole pair C , A and a left null pairp p


Ž .A , B for W oër s . Then there exists a unique matrix G such thatz z

S W s c g MM s : P z c z s 0 for each�Ž . Ž . Ž . Ž .s m=1

z in an open set containing s 4y1 npl C zI y A x q h z : x g C , h z is an m = 1Ž . Ž . Ž .p p½

ëctor- alued function analytic in an opening set containing s ,

y1and Res zI y A B h z s Gx . 2.4Ž . Ž .Ž .Ý zsz z z 50

z gs0

Ž .Here and elsewhere in this paper Res F z stands for the residue atzsz0Ž Ž .y1 .z s coefficient of z y z of a meromorphic matrix or vector func-0 0

Ž .tion F z .Proposition 2.4 can be proved adapting the argument used in Theorem

w x4.2 in 10 ; we omit the details. The matrix G in Theorem 2.4 is called thenull-pole coupling matrix or the coupling matrix for the right pole pairŽ . Ž .C , A and a left null pair A , B for W over s . We note that thep p z z

matrices in Proposition 2.4 satisfy the equality

GA y A G s B C . 2.5Ž .p z z p

Ž .This follows from the fact that S W is closed under scalar multiplications

Ž w xby meromorphic functions without poles in s see Theorem 12.2.1 in 6w x.and Theorem 3.1 in 10 .

We need one more ingredient to define the concept of spectral data. AŽ . Ž .function P z g MM V , where k F m, is called a right unit if itsk=m

w xSmith]McMillan form is I 0 . It follows from Proposition 1.1 that ifŽ . Ž .W g MM V has normal rank m y k, then there is a right unit P z gm= n

Ž .MM V such thatk=m

P z W z s 0 for all z g V . 2.6Ž . Ž . Ž .

Ž . Ž .Moreover, the right unit P z with the property 2.6 is determinedŽuniquely up to multiplication on the left by an analytic and invertible in

.V matrix function.Putting all of the above information together, we introduce the spectral

Ž .data for meromorphic matrix functions as follows. Let W g MM V , andm= nlet s : V be such that W has only finite number of poles and zeros in s .

�Ž . Ž . Ž .4 Ž .A quadruple C , A , A , B , G, P z is called a left spectral data ofp p z z


Ž .W z over s if the following properties are satisfied:

Ž . Ž . Ž .i C , A is a right pole pair of W z over s ;p p

Ž . Ž . Ž .ii A , B is a left null pair of W z over s ;z z

Ž . Ž . Ž . Ž . Ž .iii P z g MM V is a right unit such that P z W z s 0 forŽmyp.=mŽ Ž ..all z g V here p is the normal rank of W z ;

Ž . Ž .iv G is the unique matrix such that the null-pole subspace S W ofs

Ž . Ž .W z over s is given by the formula 2.4 ; i.e., G is the correspondingcoupling matrix.

Ž . Ž .If the normal rank of W z is equal to m, then P z is empty, andŽ . Ž .consequently in this case property iii above is omitted, and property iv

Ž . Ž .is reinterpreted in the sense that the condition ‘‘P z c z s 0 for each zŽ .in an open set containing s ’’ is omitted from the definition of S Ws

Ž . Ž .given in 2.4 . Furthermore, in case m s n and det W z k 0, we recoverŽ .the definition of a null-pole triple of W z ; the null-pole triples and their

connections to various interpolation and factorization problems have beenŽ .extensively studied recently, especially for rational matrix functions W z

Ž w x .see, e.g., the book 6 and references there .

Ž . Ž .THEOREM 2.5. Let W g MM V , and s : V be such that W z hasm= nfinite number of poles and zeros in s . Then there exists spectral data�Ž . Ž . Ž .4 Ž .C , A , A , B , G, P z of W z oër s . Moreoër, the spectral datap p z z

has the following additional properties:

Ž .v GA y A G s B C ;p z z p

Ž . Ž . Ž .y1vi the function P z C zI y A is analytic on s ;p p

Ž .vii let l , . . . , l be all the distinct eigenälues of A ; then the pair1 r z

A 0 ??? 0 Bz z

.0 l I . P lŽ .1 1. , .. . . .. . . .. . .� 0P l0 0 ??? l I Ž .rr

is controllable.

ŽThe following well-known lemma whose proof is provided for complete-.ness will be useful in the proof of Theorem 2.5.

Ž .LEMMA 2.6. Let A , B be pairs of matrices of sizes n = n and n = mi i i i iŽ .i s 1, . . . , r . Assume

s A l s A s B for i / j. 2.7Ž . Ž .Ž .i j


Ž . Ž . rThen the pair A s diag A , . . . , A , B s col B is controllable if and1 r i is1Ž .only if A , B is controllable for eëry i s 1, . . . , r.i i

Proof. The ‘‘only if’’ part follows immediately from the definition ofŽ Ž . .controllability the hypothesis 2.7 is not essential here . Conversely,

Ž . � 4assume that the pairs A , B are controllable. Fix i g 1, 2, . . . , r . Sincei iŽ .the pair A , B is controllable, the seti i

p A B x : p is a polynomial and x g Cm 2.8� 4Ž . Ž .i i

n i Ž .fills out the whole space C . By 2.7 , there exists a polynomial p suchiŽ .that p A has nonsingular ith diagonal block and zeros elsewhere. Hencei

the set

p A p A Bx : p is a polynomial and x g Cm 2.9� 4Ž . Ž . Ž .i

n i w ny1 xfills out 0 [ ??? [ C [ 0 [ ??? [ 0. Since the image of B AB ??? A BŽ . Ž .contains the set 2.9 , and i was arbitrary, it follows that A, B is

controllable.

Ž .Proof of Theorem 2.5. Property vi follows from the following fact: ForŽ .every p = m matrix function Q z which is analytic in a neighborhood UU

of s : V and satisfies

Q z W z s 0 for all z g V ,Ž . Ž .

the function

y1Q z C zI y AŽ . Ž .p p

Ž . Ž .is analytic in s , where C , A is a right pole pair of W z with respectp p

Ž .to s . This fact is proved in exactly the same way as Proposition 4.1 iv inw x9 .

Ž . w xProperty v has been observed in 2 ; it follows by reduction to theŽ .regular case see the remark after Proposition 2.4 .Ž .Property vii is a consequence of the definition of a left null pair. To

Ž . � 4show this, we may assume that r s 1 and s A s l . Indeed, withoutz 1Ž .loss of generality, by applying a left similarity to A , B if necessary, wez z

can assume that

r˜ ˜ Ã s diag A , . . . , A , B s col B ,Ž .ž /z 1 r z i is1


˜Ž . � 4where s A s l , i s 1, . . . , r. Now apply Lemma 2.6 withi i

˜˜ BA 0 iiA s , B s .i i0 l I P lŽ .i i

Therefore, it remains to show that the pair

BA 0 zz, , 2.10Ž .ž /0 l I P lŽ .1 1

Ž . � 4with s A s l is controllable. We may assume that A is in Jordanz 1 z

canonical form and B is constructed using the Taylor coefficients at l ofz 1� 4functions in a canonical set f , f , . . . , f of left null functions for W at1 2 k

Ž . Ž .l . Since a pair A, B is controllable if and only if a pair A y l I, B is1 1Ž .controllable, we may assume l s 0. The controllability of the pair 2.101

follows now from the fact that the matrix

f lŽ .1 1

f lŽ .2 1...

f lŽ .k 1

P lŽ .1

has linearly independent rows, which in turn follows from the orthogonal-ity of f , . . . , f at l and from the definition of left null functions W at1 k 1l .1

Using Propositions 2.1, 2.3, and 2.4, the following statement concerningthe uniqueness of spectral data is easily obtained:

Ž . Ž .THEOREM 2.7. Let W g MM V , and let s : V be such that W z hasm= na finite number of poles and zeros in s . If

C Ž j. , AŽ j. , AŽ j. , BŽ j. , GŽ j. , P Ž j. z j s 1, 2, 2.11Ž . Ž . Ž .Ž .½ 5Ž .p p z z

Ž .are two spectral data of W z oër s , then there exists an inërtible matrix SŽ . Ž .and an inërtible analytic matrix function G z z g V such that

C Ž1. s C Ž2.S, AŽ1. s Sy1AŽ2.S, P Ž2. z s G z P Ž1. z .Ž . Ž . Ž .p p p p

Ž . Ž .Here S and G z are uniquely determined by the spectral data 2.11 .Ž Ž1. Ž1.. Ž Ž2. Ž2..Moreoër, if A , B and A , B come from left null functions con-z z z z


0 l Ž Ž i..tained in the same subspace L orthogonal to W on s A , then there existsz

an inërtible matrix T such that

BŽ1. s TBŽ2. , AŽ1. s TAŽ2.Ty1 , GŽ1. s T GŽ2.S.z z z z

The matrix T is unique.

3. ADMISSIBLE QUADRUPLES

In Section 2 we saw that with each meromorphic matrix function Wdefined on V which has a finite number of poles and zeros in s : V we

�Ž . Ž . Ž .4can associate a quadruple C , A , A , B , G, P z , a spectral datap p z z

over s . In this section we introduce the abstract concept of quadruples, bytaking the properties of the spectral data which do not depend on thefunction W as part of the definition. Thus, a notion of quadruples emerges,which a priori are not connected to any particular function, but neverthe-less potentially can serve as a spectral data for some W. Finding such W isthe basic interpolation problem which will be solved in the next section.

DEFINITION 3.1. Let C g Cm= np , A g Cnp=np , A g Cnz=nz , B gp p z znz=m nz=np Ž . Ž .C , and G g C be matrices, let P z g MM V , and let s : V.k=m

�Ž . Ž . Ž .4We will call C , A , A , B , G, P z an admissible quadruple over s ifp p z z

Ž . Ž . Ž .i the pair C , A is observable and s A : s ,p p p

Ž . Ž . Ž .i the pair A , B is controllable and s A : s ,z z z

Ž . Ž .iii P z is a right unit,Ž .iv GA y A G s B C ,p z z p

Ž . Ž . Ž .y1v the function P z C z y A is analytic on s ,p p

Ž .vi the pair

bA zz

P lŽ .l I 0 11

l I P lŽ .,2 2. .0 . .. .� 0

l I P lŽ .r r

is controllable, where l , l , . . . , l are all distinct eigenvalues of A .1 2 r z

�Ž . Ž . Ž .4PROPOSITION 3.1. Suppose t s C , A , A , B , G , P z is an ad-i p p z z i ii i i iŽ . Ž . Ž . Ž .missible quadruple oër s : V i s 1, 2 , and P z s Q z P z for somei 1 2


Ž .unit Q g MM V . If s l s s B, thenk=k 1 2

A 0 A 0 Bp z z1 1 1C Ct s , , , ,p p1 2½ 0 A 0 A Bž / ž /p z z2 2 2

G G1 12 , P zŽ .1 5G G21 2

is an admissible quadruple oër s j s where G and G are the unique1 2 12 21matrices which satisfy Lyapuno equations

G A y A G s B C ; G A y A G s B C .12 p z 12 z p 21 p z 21 z p2 1 1 2 1 2 2 1

Ž . Ž .It is straightforward to verify for t conditions i ] vi in Definition 3.1,therefore we omit the details of the proof. The quadruple t in Proposition3.1 is called the union of the quadruples t and t . A quadruple t is1 2 i

Ž .called a s -restriction of t i s 1, 2 .iIf t and t are admissible quadruples representing the spectral data of1 2

Ž .W g MM V over s and s , respectively, where s g s s B, then them= n 1 2 1 2union of t and t is the spectral data of W over s j s .1 2 1 2

�Ž . Ž . Ž .4PROPOSITION 3.2. Suppose t s C , A , A , B , G, P z is an ad-p p z z

missible quadruple oër s . Let S and T be nonsingular matrices such that

A )p 1y1 C CC S, S A S s , ,p pŽ . 1 2p p 0 Až /p 2

A 0 B G Gz z1 1 11 12y1TA T , TB s , , T GS s .Ž .z z ) A B G Gž /z z 21 222 2

�Ž . Ž . Ž .4Then the quadruple t s C , A , A , B , G , P z is also an admissi-1 p p z z 111 1 1 1

ble quadruple oër s .

The quadruple t in Proposition 3.2 is called a restriction of the1quadruple t , and t is called an extension of t .1

4. BASIC INTERPOLATION PROBLEM

We are given an open connected domain V : C. At each point l g V� 4an admissible quadruple over l

p s C , A , A , B , G , P z 4.1Ž . Ž .� 4Ž . Ž .l p p z z l ll l l l


Ž Ž . Ž .is given the case when at least one of the pairs C , A and A , B isp p z zl l l l

. Ž .empty is not excluded; in this case G is empty as well . Here P z isl l

Ž .considered as an element of MM V , where V : V is an open neigh-k=m l l

borhood of l. The problem is to determine when there exists a meromor-phic matrix function W with the spectral data at each l given by t , and inl

such case to construct W.The first obvious necessary condition is

Ž . Ž .A The set of points l such that t contains nonempty pair C , A ,l p pl l

Ž . Ž .or A , B , is discrete no accumulation points in V .z zl l

Ž .Consider A has to be satisfied in order for W to be meromorphic in V.Another necessary condition is

Ž . Ž . Ž .B If the domains V and V of P z and P z , respecti ely, haël l l l1 2 1 2Ž .a nonempty intersection D, then there exists a matrix function W z suchl l1 2

Ž . w Ž .xy1that W z and W z are analytic on D andl l l l1 2 1 2

P z ' W z P z . 4.2Ž . Ž . Ž . Ž .l l l l1 1 2 2

Ž .Indeed, the equality 4.2 follows from the fact that the rows of P andl1Žthe rows of P span the same vector subspace over the field of scalarl2

.meromorphic functions defined on D ; namely, the subspace of meromor-Ž .phic row functions on D annihilating W. Therefore, 4.2 holds with Wl l1 2

and Wy1 meromorphic on D; but since both P and P are analytic on D,l l l l1 2 1 2y1 Žthe functions W and W are analytic as well use the Smith]McMillanl l l l1 2 1 2.forms for P and P to verify that .l l1 2

Ž . Ž .Conditions A and B turn out to be also sufficient:

THEOREM 4.1. Suppose that for eëry l g V, an admissible quadruple tl

� 4 Ž . Ž .oër l is gi en, and assume that conditions A and B aboë are satisfied.Then there exists a meromorphic matrix function W on V whose spectral dataat each l is equal to t . Moreoër, such W can be chosen so that the numberl

of columns of W is equal to its normal rank.The proof of Theorem 4.1 will be given by an explicit construction of the

interpolant W, in several steps.

Ž .Proof. First, of all, observe that in view of 4.2 , we have

W z ' W z W z 4.3Ž . Ž . Ž . Ž .l l l l l l1 3 1 2 2 3

w xfor all z g V l V l V . Thus, in the terminology of 23, 24 , thel l l1 2 3� Ž . 4collection of functions W z : l , l g V forms a cocycle. It followsl l 1 21 2

Ž w x.from general results on fiber bundles see, e.g., 17 that every cocycle ofinvertible matrix functions is trivial. A direct proof of triviality of cocyclesin an open subgroup of invertible elements in a Banach algebra is given inw x w x24 ; see also Theorem 1.2 in 23 , where a version of this result is proved


for a compact V. Therefore, there exists a collection of analytic and� Ž .4 Ž .invertible matrix functions V z , where V z is defined on V , such thatl l l

y1W z s V z V z , z g V l V . 4.4Ž . Ž . Ž . Ž .l l l l l l1 2 1 2 1 2

Define the function

P z s V z P z , z g V .Ž . Ž . Ž .l l l

Ž . Ž . Ž . Ž Ž .Because of 4.2 and 4.4 , the function P z is well defined i.e., P z is.independent of the choice of l s l or l s l if z g V l V ; more-1 2 l l1 2

Ž . Ž Ž ..over, P z is analytic in V, and is right unit as an element of MM V .k=mŽ . Ž . Ž .We can and will therefore replace P z by P z in the definition of t .l l

Ž .We now construct W l step by step.

Step 1. Let K be the discrete set of all l g V such that at least oneŽ . Ž .of the pairs C , A and A , B is nonempty. For every l g K there isp p z zl l l l

Ž . Ž .an m = m meromorphic even rational matrix function R z such thatl

Ž . �Ž . Ž . 4 Ž .det R z 'u 0 and C , A , A , B , G is the spectral data of R z atl p p z z l ll l l l

Ž . w xl. An explicit construction of R z is given in 21 , see also Section 4.6 inl

w x Ž .6 . Write the Laurent series for R z centered at l,l

`jR z s z y l R ,Ž . Ž .Ýl l j

jsp

where the integer p depends on l. It follows from the definition of theŽ .spectral data that there exists an integer q G p depending on l with the

Ž .following property: Every meromorphic in a neighborhood of l m = mŽ .matrix function V z with the Laurent series of the form

q `j j ˜V z s z y l R q z y l R ,Ž . Ž . Ž .Ý Ýl j l j

jsp jsqq1

˜where R are arbitrary, has determinant not identically zero and thel j�Ž . Ž . 4 w xspectral data C , A , A , B , G at l. By Theorem 3.2 of 25 therep p z z ll l l l

Ž .exists an m = m meromorphic in V function W , with the following1properties:

Ž . Ž .i det W z k 0;1

Ž . Ž .ii K is the singular set of W z ;1

Ž . �Ž . Ž . 4iii for every l g K, the quadruple C , A , A , B , G is thep p z z ll l l l

spectral data of W at l.1

Step 2. Using Proposition 1.1, write W s EDF, a Smith]McMillan1Ž . Ž Ž . Ž ..factorization of W , and let W s ED. Here D z s diag d z , . . . , d z1 2 1 m


Ž . Ž . Ž .where d z i s 1, . . . , m are meromorphic in V scalar functions suchithat d dy1 are analytic for i s 1, . . . , m y 1.iq1 i

Observe that W and W have the same singular set K. Moreover, since1 2Ž .multiplication on the right by an analytic and invertible on V matrix

function does not change the spectral data, W has the same spectral data2�Ž . Ž . 4C , A , A , B , G at every l g K as W .p p z z l 1l l l l

After applying, if necessary, a similarity transformation to each�Ž . Ž . 4 Ž .C , A , A , B , G ; l g K, we may assume that A , B has beenp p z z l z zl l l l l l

Ž .y1constructed from the bottom k rows of E z , where k is the geometricl l

Ž .multiplicity of the zero of W or of W at l. This assumption will be used1 2in Step 4 below.

Step 3. Let h be the largest sum of the geometric multiplicity of apole and the geometric multiplicity of a zero of W in V. Observe that2h F m. We construct now a ‘‘tall’’ matrix function W with h columns and3m rows. Let m be the largest geometric multiplicity of a zero of W in V.2

Ž .Define the m = h matrix function Q by the blank positions are zeros

1. . .

1q1

.Q s . .1 qm

. . .1

with q ’s analytic functions in V having the following property. Suppose qi iis in the jth row of Q, and k s j q m y h, so that the nonzero entries qiand 1 of the jth column of Q, are in positions j and k, respectively. Pickl g V and suppose the Laurent expansions of d and d at l arej k

` `l lz y l d and z y l d ,Ž . Ž .Ý Ýj , l k , l

lsp lspj k

where d / 0 and d / 0. If p F 0, q has a nonzero value at l. Ifj, p k , p k ij k

p ) 0, q has a zero at l of multiplicity p y p . Note that such q exists byk i k j iw xTheorem 3.1 in 27 .

Let W s W Q. Note that by the choice of h, the functions W and W3 2 3 2have the same poles with the same multiplicities in V. Also, the jthcolumn of W vanishes at a point l g V to the order k if and only if the3jth column of W vanishes at l to the same order k.2


Ž .Step 4. In this step we project the columns of W onto ker P z along3� Ž .4a subspace which we proceed to construct. Let r z be a collectionl lg K

Ž .of scalar functions r z that are analytic in V and have the followingl

properties:

Ž . Ž . Ž . � 4i r l s 1; r z s 0 for z g K _ l ; and for every l g K wel l 0 0 0Žq.Ž . Ž . Žhave r l s 0 for q s 1, . . . , k q k , where k s k l resp. k sl 0 z p z z 0 p

Ž .. Ž .k l is the largest zero resp. pole multiplicity of W at l .p 0 1 0

Ž . < Ž . <ii The series Ý r z converges uniformly on compact subsetslg K l

of V.

Ž . Ž . Ž .The proof of existence of r z satisfying i and ii is given in thel

Appendix.Ž .Property ii guarantees that for every choice of a bounded collection of

� 4 Ž .p = q matrices U , the series Ý r z U converges in V andl lg K lg K l l

Ž .represents an analytic in V matrix function.Ž . Ž .y1Let e z be the ith row from the bottom of E z , i s 1, 2, . . . , m,i

where m is the largest geometric multiplicity of a zero of W in V as inStep 3. It follows from the admissibility of the quadruple t that the matrixl

e lŽ .kl

...Q l [Ž .e lŽ .1

P lŽ .

Žhas full row rank here k is the geometric multiplicity of the zero of W atl 1.l , at each l g K. Also,

1 0col e z E l s ,Ž . Ž .Ž . iski l I

and, if W is the matrix whose columns are the leading coefficients in the3l

Laurent expansions at l of the columns of W , the product3

1col e l WŽ .Ž . iski 3ll

Ž . Žis nonsingular. Hence there exists an m y k = m matrix e where k isl

.the number of rows of P with bottom k rows equal to 0 such thatl

rank G W s rank W ,Ž .l 3l 3l


where

01G s e q ,l l col e lŽ .Ž .j jsm

and the matrix

Gl

P lŽ .

Ž . 5 5is nonsingular. We may and will assume e F 1. Letl

0e zŽ .m

F zŽ .. e 1l.F z s q r r [ .Ž . Ž .Ý. l P zŽ .0lgKe zŽ .1

P zŽ .

Ž . Ž .Note that F z is m = m and F l is invertible for all l g K. Now let

F zŽ .y1 1W z s F z W z .Ž . Ž . Ž .4 30

By construction, the discrete data of W are the same as those of W for4 3every l g K, but W may have additional spectral points not in K. Also4

Ž .F zy1 1Ž . Ž .note that F z is a projection onto the right annihilator of P z .0

Ž .Step 5. We continue to use the notation introduced in Step 4. It will� 4 Žbe convenient to represent K as a denumerable set K s l , l , . . . if K1 2

.is finite or empty, the changes in the subsequent arguments are obvious .Ž .Let Q j s 1, 2, . . . be the compact sets taken from Lemma 6.2 in thej

Ž .Appendix. For each j s 1, 2, . . . choose an analytic on V matrix functionŽ .F z with the following properties:l j

Ž . 5 Ž .5i F z F 1 for z g Q ;l jj

Ž . Ž . Ž .ii P l F l s 0;j l jj

Ž . Ž .iii the column span of F z is orthogonal at l to the columnl jj

span of W ;4

Ž . w Ž . Ž .T xiv the matrix W z F P z is of square size;4 l Ž z .j

Ž . Ž . Ž . Ž . Ž .v e z F z vanishes at l to the order k whenever e z W zi l j i 4j

vanishes at l to the order k.j


Ž .The existence of such F z can be proved analogously to the proof ofl jŽ . Ž . Ž . Ž .existence of u z satisfying 6.1 and 6.2 see the Appendix . Letj

F zŽ .y1 1W z s F z r z F z ,Ž . Ž . Ž . Ž .Ý4 l lž /0 lgK

and let

˜W z s W z W z .Ž . Ž . Ž .5 4 4

Note that the function W has he same zeros and poles at the spectral5Ž .points of W as W , and the column span of W fills out ker P z .1 4 5

Ž . Ž . ŽStep 6. Let D z be the Smith]McMillan form of W z over V see5 5.Proposition 1.1 ; so D is diagonal, and W s E D F , where E and F5 5 5 5 5 5 5

are analytic and invertible in V. Let

˜W s E D D ,5 5

˜where D is a diagonal meromorphic matrix function without zeros andpoles at the spectral points of W and such that all zeros and poles of W1

ãre only at the spectral points of W . Existence of such D is easily reduced1to the existence of a scalar meromorphic function with given poles and

Ž .given zeros including the multiplicity of each pole and zero ; this in turn isw xensured by Theorem 3.1 in 27 .�Ž . Ž . Ž .4We have to show that t s C , A , A , B , G , P z is the spectrall p p z z l ll l l l

data of the constructed W at l for each l g V. Also, it suffices toconsider only the points l g V for which at least one of the pairsŽ . Ž .C , A and A , B is nonvacuous. Fix such a point l. Thenp p z zl l l l

S W s S WŽ . Ž .l 2 l 1

y1 npls C zI y A x q h z : x g C , h z is an m = 1Ž . Ž .Ž .½ p pl l

vector-valued function analytic at l and

y1Res zI y A B h z s G x . 4.5Ž . Ž .Ž . 5zsl z z ll l

Ž . Ž . Ž .Clearly, S W : S W and C , A is a right pole pair for W at l.l 3 l 2 p p 3l l

Since the columns of W are orthogonal at l, and the ith column from the3right of W vanishes at l to the order k if and only if the ith column3 ifrom the right of W vanishes at l to the order k , the partial multiplici-2 ities of l as the zero of W and as the zero of W coincide. Also, e W3 2 i 3Ž Ž .y1 .where e is the ith row from the bottom of E z vanishes at l to theiorder k if and only if e W vanishes at l to the order k , and the lineari i 2 i


span of the leading coefficients in the Laurent expansion of the columns ofW at l intersects trivially with the subspace3

kker col e l ,Ž .Ž .j js1

where k is the geometric multiplicity of the zero of W at l. It follows that2� 4e , . . . , e is a canonical set of left null functions for W at l, and1 k 3Ž . w xA , B is a left null pair for W at l. By Lemma 3.2 in 10 ,z z 3l l

�Ž . Ž . Ž .4 Ž .C , A , A , B , G , P z with P z a right unit is a spectral datap p z z l 3l 3ll l l l

for W at l. Thus,3

S W s ker P l T ,Ž .l 3 3l l

Ž . Ž .where T is the right-hand side of 4.5 . The function F z constructed inl

Step 4 takes nonsingular value at l, so the projection performed in Step 4Ž .is analytic. Since P z W is analytic at l, the projection does not effect the3

Ž .right pole pair C , A . Also, each column of W differs from thep p 4l l

corresponding column of W by a function f such that e f s 0 for each3 ipositive integer i less than or equal to the geometric multiplicity of the

Ž .y1 Ž .zero of W at l. Hence zI y A B f z is analytic at l and it follows3 z z

that

S W s ker P l TŽ .l 4 4l l

for some right unit P .4l˜ Ž .Since W z has neither zero nor a pole at l, and the column spans of4˜W and W are orthogonal at l, spectral data of W and of W differ in the4 4 4 5

�Ž . Ž . Ž .4annihilator function and C , A , A , B , G , P z is a spectral datap p z z ll l l l

for W at l. This data is not affected by Step 6.5

The function W constructed in Theorem 4.1 has linearly independentŽ .columns over the field of scalar meromorphic functions on V . It turns

out that it is unique up to multiplication on the right by a unit.

Ž .THEOREM 4.2. Suppose W and W are functions in MM V with a1 2 m=n�Ž . Ž . Ž .4spectral data at each point l g V equal to C , A , A , B , G , P zp p z z l ll l l l

Ž . Ž . Ž .with P z g MM V . Then W s W Q for some unit Q g MM V .l Žmyn.=m 2 1 n=n

Proof. Let E D F be a Smith]McMillan factorization of the functioni i iŽ . Ž .y1W i s 1, 2 , and let E be a function formed by the top n rows of E z .i 1

Then EW and EW are two regular meromorphic n = n matrix functions1 2�Ž . Ž . 4with a spectral data C , A , A , B , G at each point l g V, so thep p z z ll l l lw xresult follows from Corollary 2.9 of 4 .


5. MINIMAL DIVISIBILITY

ŽIn this section we characterize minimal divisibility suitably understood,.as it is explained below of rectangular meromorphic matrix functions in

terms of spectral data. More exactly, classes of minimal divisors will bedescribed in terms of restrictions of spectral data of the dividend functionŽ .see Theorem 5.4 below .

Ž .Let V be a domain in the complex plane, and let W g MM V . Wem= nconsider factorizations of the form

W s W W . 5.1Ž .1 2

Ž . Ž .where W g MM V and W g MM V , together with the condition1 m=k 2 k=n

d W s d W q d W , 5.2Ž . Ž . Ž . Ž .l l 1 l 2

Ž . Ž .where d X denotes the sum of the pole multiplicities if any of X at l.l

ŽSeveral definitions of a minimal factorization in the framework of. w xrational matrix-valued functions have been used in the literature. In 18 ,

Ž . Ž .the factorization 5.1 is called minimal at l if 5.2 holds and W, W , W1 2w x Ž . Ž .are all square. In 33 , a factorization 5.1 is called minimal at l if 5.2

w xholds and the functions involved have arbitrary sizes. In 34 , the factoriza-Ž . Ž . Ž .tion 5.1 is called minimal at l if 5.2 holds and W resp. W has full1 2

Ž .column resp. row rank. The necessary and sufficient conditions forexistence of a minimal factorization according to each definition are

w x Žknown for rational matrix functions 18, 33, 34 . The regular case i.e., W1.and W are square size with determinant not identically zero of minimal2w xfactorizations of rational matrix functions has been studied in 16, 15, 35 ,

w xwhile the nonregular case has been studied in 18, 34, 29 .w xIn this section we adopt the concept of minimal divisibility as in 34 . If

Ž . Ž .5.1 and 5.2 hold, and k is the normal rank of W, we say that theŽ .factorization 5.1 is minimal at l. W is called a minimal left di isor of W1

Ž . Žon s if the factorization 5.1 is minimal at every l g s here s is a.subset of V .

All the concepts introduced above become especially manageable whenV is the complex plane and W is a rational matrix function without a poleor zero at infinity. In this case the spectral data can be read off directlyfrom a realization. Suppose W is an m = n rational matrix function

Ž .without a pole or zero at infinity. Let A, B, C, D be a minimal realizationof W, i.e.,

y1W z s D q C zI y A B ,Ž . Ž .

with the size A as small as possible. Since W does not have a zero at


infinity, the rank of D is equal to the normal rank of W. If D‡ is ageneralized inverse of D, i.e., DD‡D s D and D‡DD‡ s D‡, then the leftkernel of W is spanned by the rows of I y WW >, where

y1> ‡ ‡ ‡ ‡W z s D y D C zI y A q BD C BDŽ . Ž .

Ž w x.see 29 . Similarly, the right kernel of W is spanned by the columns ofI y W >W. A left kernel polynomial of W is a matrix polynomial whose

w xrows form a minimal polynomial basis 19 for the left kernel of W.Let s : C, let S be a finite set containing all the poles and zeros of W,

‡ Žand let s s S l s . Pick a generalized inverse D and D whose row resp.ˆ. Ž .column span is orthogonal to the left resp. right kernel of W on S, and

let A>s A y BD‡C. Pick nonsingular matrices S and T such thatŽŽ y1 . Ž > y1 ‡. .CS, S AS , TA T , TBD , TS equals

A 0 BA 0 z z G Gp 11 12C , C , , , , ,p p ž /G Gž /0 A 0 A Až /� 021 22p z z

Ž . Ž . Ž Ž . Ž ..where s A j s A : s , s A j s A l s s B, and the size ofˆ ˆp z p z

w xG is such that A G A is defined. Then, by Theorem 3.1 in 30 ,11 z 11 p

�Ž . Ž . 4 Ž .C , A , A , B , G is a left null-pole triple for W over s , andp p z z 11�Ž . Ž . Ž .4C , A , A , B , G , P z is the spectral data of W over s .p p z z 11

Let W, W , and W be rational matrix functions without zeros or poles1 2at infinity, and suppose factorization W s W W is minimal at each point1 2of the complex plane. Then fW s 0 if and only if fW s 0, so the left1

Ž .kernel polynomials of W and W coincide. Let A , B , C , D be a mini-1 i i i iL Ž R.mal realization of W , i s 1, 2. Pick a one-sided inverse D resp. D ofi 1 2

Ž . Ž .D resp. D which is orthogonal to the left resp. right kernel of W on1 2the set s containing all poles and zeros of W. Since the factorizationˆW W is minimal,1 2

A B C B D1 1 2 1 2 C D C, , , D D1 1 2 1 2ž /0 A B2 2

is a minimal realization of W. If D‡ s D RD L, then2 1

A B C B D1 1 2 1 2> ‡ C D CA s y D 1 1 20 A B2 2

LA y B D C 01 1 1 1s .R) A y B D C2 2 2 2


If T is a nonsingular matrix such that1

ˆ Â 0 B1 1L y1 LT A y B D C T s and T B D s ,Ž .1 1 1 1 1 1 1 1 10 A B1 1

ˆ ˆ ˆŽ . Ž . �Ž . Ž . 4with s A : s and s A l s s f, then t s C , A , A , B , Gˆ ˆ1 1 1 1 1 1 1 11with G an upper block of T is a global left null-pole triple for W . On11 1 1the other hand, if T is a nonsingular matrix such that

LB Dˆ ˆ1 1A 0 B> y1TA T s and T s ,‡B D0 A B2

ˆ ˆ ˆŽ . Ž . �Ž . Ž . 4with s A : s and s A l s s B, then t s C, A , A, B , G with Gˆ ân upper block of T is a global null-pole triple of W. Since one can take

I 0 T 02 1T sT T 0 I3 2 1

ˆwith identity I of the same size as A and suitable matrices T and T , it2 1 2 3follows that t is a restriction of t .1

We now return to the general framework of meromorphic matrix func-tions. If V : C, a necessary and sufficient condition for being a minimal

w xleft divisor has been obtained in 2 ; it generalizes the observation made inthe previous paragraph for rational matrix functions:

Ž . Ž .THEOREM 5.1. W g MM V is a minimal left di isor of W g MM V1 m=k m=noër s , where k is the normal rank of W, if and only if the spectral data of W1oër s is a restriction of the spectral data of W oër s .

w xTheorem 5.1 has been proved in 2, Theorem 4.1 by reducing theproblem to divisibility of meromorphic square matrix-valued functions, and

w xthen using the results from 4 . In view of Theorem 4.1 above, Theorem 5.1has the following corollary.

COROLLARY 5.2. Suppose W and W are meromorphic matrix functions1on V : C haïng the same number of rows. Then there exists a meromorphic

Ž .matrix function W such that W s W W and 5.2 holds for each l g V if2 1 2and only if the spectral data of W oër V are a restriction of the spectral data1of W oër V.

˜Proof. By Theorem 4.1, there exists a meromorphic matrix function W1whose number of columns equals the normal rank of W and which has thesame spectral data as W . By Theorem 5.1, there exists a meromorphic1

˜ ˜ ˜matrix function W such that W s W W is a minimal factorization. Also,2 1 2


by Theorem 5.1 there exists a matrix function M analytic on V such that˜W s W M. By Proposition 2.4, for each l g V there exists a matrix1 1

˜ Ž .function M analytic in a neighborhood of l such that W z sl 1Ž . Ž . Ž . Ž .W z M z . Then, for each l g V, M z M z s I in a neighborhood of1 l l

l, and it follows from Proposition 1.1 that M is a right unit. Hence

R ˜W s W M W \ W W ,Ž .1 2 1 2

R Ž .with M a right inverse of M which is analytic in V, and 5.2 holds.

ŽWe note that the assumption V : C can be replaced after extending.the definition of spectral data to include the point at infinity by the

� 4assumption that V be a proper subset of C j ` . As the next example� 4shows, Corollary 5.2 is no longer valid when V s C j ` .

Ž . w Ž .y2 xEXAMPLE 5.3. Let W z be the 1 = 2 matrix function 1, z z y 1 ,Ž . w Ž .y2 xand let W z s 1, z y 1 . Then W and W have the same global left1 1

null-pole triple

1 1w x1 1 , , B, B .½ 5ž /0 1

Ž .One can verify that there exists no constant matrix S such that W z sŽ .W z S.1

Theorem 4.1 allows us also to obtain additional information aboutminimal left divisors, namely, classify them by the right equivalencerelation. Two minimal left divisors W and W of W are called s-right11 12

Ž .equi alent if the equality W s W X holds for some necessarily unique11 12Ž .k = k matrix function X z which is analytic on an open neighborhood of

s and invertible for all z g s .

Ž .THEOREM 5.4. Let W g MM V be a function with normal rank k.m= nThere is a one-to-one correspondence between the restrictions of the spectraldata of W oër s and the classes Q of s-right equi alent minimal left di isorsof W.

Proof. This correspondence is constructed as follows: Given a restric-Ž .tion t of the spectral data of W over s , let W g MM V be the1 m=k

meromorphic matrix function constructed in the proof of Theorem 4.1.Then the class Q corresponding to t is defined by requirement that W is1a representative of Q.

Conversely, let be given a class Q. Choose a representative W of Q1such that the number of columns in W is equal to k, the normal rank of1W. The existence of such representative is ensured by Theorem 4.1. ByTheorem 5.1 the spectral data t of W over s is a restriction of the1


spectral data of W over s . The same Theorem 5.1 ensures that t isindependent of the choice of the representative W in Q.1

6. APPENDIX

Ž .In this appendix we prove the existence of functions r z which arel

Ž . Ž .analytic in V and satisfy i and ii of Step 4 in Section 4. In fact, we provea slightly more general result.

THEOREM 6.1. Let V : C be a domain, and let K : V be a discrete set,� Ž .4i.e., without limit points in V. Then there is a collection r z of scalarl lg K

Ž .functions r z that are analytic in V and haë the following properties:l

Ž . Ž . Ž . � 4i r 1 s 1; r z s 0 for z g K _ l ; and for eëry l g K wel l 0 0 0Žq.Ž . Ž .haë r l s 0 for q s 1, . . . , k, where the positi e integer k s k l, ll 0 0

depends on l and l .0

Ž . < Ž . <ii The series Ý r z conërges uniformly on compact subsetslg K l

of V.

We need some preliminaries. A set M : C is called finitely connected ifM is connected and C _ M consists of a finite number of connectedcomponents. A set N : M is called simply connected relative to M if for

Ž .every connected component Y of C _ N the set Y l C _ M is nonempty.

LEMMA 6.2. Let V : C be a domain. There exists a nondecreasingsequence Q : Q : ??? of compact sets with the following properties:1 2

Ž .i Each Q is finitely connected and simply connected with respectito V;

Ž . ìi D Q s V;is1 i

Ž .iii for eëry compact set Q : V there is index p such that Q : Q .p

w xA proof of Lemma 6.2 can be found, for example, in 26, Lemma 9.2 .

Ž . Ž .Proof of Theorem 6.1. The existence of r z satisfying i is a standardl

Ž w x.fact in the theory of analytic functions see, e.g., Theorem 15.15 in 32 .Without loss of generality we can assume that K is an infinite set

� 4K s l ,l , . . . and that l f Q for sufficiently large p, where Q are1 2 pq1 p pŽ . Ž . Ž .taken from Lemma 6.2. If r z are analytic functions on V satisfying i ,l j

Ž . Ž . Ž . Ž .we replace each r z by u z r z , where u z is an analytic functionl j l jj

such that

u l s 1; uŽq. l s 0 for q s 1, . . . , k l , l , 6.1Ž .Ž . Ž . Ž .j j j j j j


and

y12u z F j max r z , z g Q . 6.2Ž . Ž . Ž .j l jy1ž /jzgQ jy1

Ž . Ž . Ž . Ž .The property 6.1 ensures that i remains valid for u z r z , andj l jŽ .because of 6.2 we have

y2u z r z F j , z g Q ,Ž . Ž .j l jy1j

Ž . Ž . Ž .and therefore ii holds as well for u z r z .j l jŽ . Ž . Ž .It remains to verify the existence of u z satisfying 6.1 and 6.2 , atj

Ž .least for sufficiently large j. We seek u z in the formj

su z s 1 q z y l ¨ z ,Ž . Ž .Ž .j j j

Ž . Ž .where s s k l , l q 1, and where ¨ z is analytic in V and such thatj j j

ys¨ z q z y l - e , z g Q .Ž . Ž .j j jy1

Ž .Here e ) 0 is chosen sufficiently small so that 6.2 holds. Since l f Q ,j jy1Ž .the existence of ¨ z with the requisite properties is ensured by a suitablej

Ž w x.approximation theorem for example, Lemma 18.5.1 in 22 .

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Interpolation and Divisibility of Meromorphic Matrix Functions · rational matrix functions; numerous interpolation problems for this class of functions and their various applications,